#### Transcript of Exponents- Bacterial Growth

**Exponents- Bacterial Growth**

Results

After only five seconds we have 32 bacteria. Maybe not a lot but quite a few when you imagine that you only started with one!

What would happen if we extended the time? Instead of a five-second rule, what about a 30-second one? A minute? An hour? We need to find a mathematical pattern instead of just extending our line

Step 1

Since our situation is doubling, we can look at what patterns in math follow this. If you look at how the numbers are increasing, they do not follow a pattern of addition. It is instead increasing by multiplication if you multiply number of bacteria before by two you get the current amount of bacteria. Since we are repeating multiplying by two, we can think of exponents with a base of two since that is the number we keep on multiplying. The exponent is the amount of time which tells us how many times we have to multiply two by itself.

Step 2

n our example, after five seconds we have 2 x 2 x 2 x 2 x 2 bacteria which equals 32. Written another way, we say 25=32.

Let's try our 30 second example. The base remains 2 and the exponent is 30 or 230. Solving we get 1,073,741,824! That is an incredible amount of bacteria for only 30 seconds.

Amount Now

For one hour we solve for 23600 (one hour = 60 minutes = 60 x 60 = 3600 seconds) to get our answer in scientific notation:

5.10 x 101083. This number is actually more than the amount of atoms in the known universe which is estimated at 1078 to 1081!

Introduction

The five second rule states that if food falls to the floor, you can still safely eat it as long as you pick it up in five seconds. But as the cartoon on your left shows, does bacteria really pay attention to five seconds? How much time does it take for bacteria to multiply in that time?

Exponential Growth

The picture at left shows an example of exponential growth. As you can see, the growth starts off slowly on the left of the graph but then rises quickly and this seems to explode as you move further right.

How can math help us with this rule? Let's say that we have a bacteria that doubles every second. We can set up a graph indicating how much bacteria we be on our food after five seconds.

Zero Seconds

1 Bacteria

1 Seconds

2 Bacteria

2 Seconds

4 Bacteria

3 Seconds

8 Bacteria

4 Seconds

16 Bacteria

5 seconds

**Bacteria Growth**

32 Bacteria

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